A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types

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space-time domain decomposition for two-phase flow between different rock types

Elyes Ahmed, Sarah Ali Hassan, Caroline Japhet, Michel Kern, Martin Vohralík

To cite this version:

Elyes Ahmed, Sarah Ali Hassan, Caroline Japhet, Michel Kern, Martin Vohralík. A posteriori error

estimates and stopping criteria for space-time domain decomposition for two-phase flow between differ-

ent rock types. SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées

et Industrielles (SMAI), 2019, 5, pp.195-227. �10.5802/smai-jcm.47�. �hal-01540956v3�

for space-time domain decomposition

for two-phase flow between different rock types ^∗

Elyes Ahmed ^†‡§ Sarah Ali Hassan ^† Caroline Japhet ^‡ Michel Kern ^† Martin Vohral´ık ^†

Abstract

We consider two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. This is a nonlinear and degenerate parabolic problem with nonlinear and discontinuous transmission conditions on the interface.

We first describe a space-time domain decomposition method based on the optimized Schwarz waveform relaxation algorithm (OSWR) with Robin or Ventcell transmission conditions. Complete numerical approximation is achieved by a finite volume scheme in space and the backward Euler scheme in time.

We then derive a guaranteed and fully computable a posteriori error estimate that in particular takes into account the domain decomposition error. Precisely, at each iteration of the OSWR algorithm and at each linearization step, the estimate delivers a guaranteed upper bound on the error between the exact and the approximate solution. Furthermore, to make the algorithm efficient, the different error components given by the spatial discretization, the temporal discretization, the linearization, and the domain decomposition are distinguished. These ingredients are then used to design a stopping criterion for the OSWR algorithm as well as for the linearization iterations, which together lead to important computational savings. Numerical experiments illustrate the efficiency of our estimates and the performance of the OSWR algorithm with adaptive stopping criteria on a model problem in three space dimensions. Additionally, the results show how a posteriori error estimates can help determine the free Robin or Ventcell parameters.

Key words: two-phase Darcy flow, discontinuous capillary pressure, finite volume scheme, domain de- composition method, optimized Schwarz waveform relaxation, Robin and Ventcell transmission conditions, linearization, a posteriori error estimate, stopping criteria

1 Introduction

Two-phase flows in porous media are of interest in many applications, such as CO2 sequestration in saline aquifers, description of oil reservoirs, or gas migration around a nuclear waste repository in the subsurface.

The numerical simulation of such flows is a challenging task. One well-known reason, which is the main topic of this paper, is that the capillary pressure and the relative permeability functions may be discontinuous across the interface between different regions of the domain.

We consider in this paper a simplified two-phase flow model (one equation, no advection) introduced in [24] to study the phenomenon of oil or gas trapping in a porous medium with several rock types (see

∗ This work was supported by the French ANR DEDALES under grant ANR-14-CE23-0005, Labex MME-DII, and the French Agency for Nuclear Waste Management (ANDRA). It also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 647134 GATIPOR).

† Inria, 2 rue Simone Iff, 75589 Paris, France & Universit´ e Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vall´ ee, France.

michel.kern@inria.fr, martin.vohralik@inria.fr

‡ Universit´ e Paris 13, Sorbonne Paris Cit´ e, LAGA, CNRS UMR 7539, 93430, Villetaneuse, France.

japhet@math.univ-paris13.fr

§ Department of Mathematics, University of Bergen, Bergen, Norway. elyes.ahmed@uib.no

§ UPMC Sorbonne Universit´ e, Laboratoire Jacques-Louis Lions, B.C. 187, Paris, F-75005 France.

sarah.alihassan.upmc@gmail.com

1

also [14, 52] for the analysis of a 1D model). The main theoretical and numerical difficulties of this problem are the nonlinearity and degeneracy of the parabolic equation for the saturation of one of the phases, together with nonlinear and discontinuous transmission conditions at the interface between the rocks. Existence of a weak solution has been proven in [24], using the convergence of a finite volume scheme (see also [17, 18], and [15] for the full two-phase flow model). Uniqueness of a weak solution has been proven in [17] for a particular choice of functions characterizing the porous medium and in [18, 19] for the full two-phase problem with advection terms in the one-dimensional case. An a priori convergence study of a finite volume discretization has been undertaken in [24]. Due to different hydrogeological properties of the different rocks, domain decomposition (DD) methods appear to be a natural way to solve efficiently two-phase flow problems, and are proposed in [55, 56] as well as [31, 39, 48, 49, 50, 51].

The first purpose of this paper is to propose a domain decomposition method that is global-in-time, with time-dependent, nonlinear, and discontinuous optimized Ventcell transmission conditions at the interface between rock types, for solving the two-phase flow model of [24]. This method is based on the optimized Schwarz waveform relaxation algorithm (OSWR), in which at each OSWR iteration, space-time nonlinear subdomain problems over the whole time interval are solved. The exchange between the subdomains is here ensured using the nonlinear Ventcell transmission operators. Crucially, this approach allows for different time stepping in different parts of the domain, adapted to the physical properties of each subdomain, and for parallelization in time, in contrast to the domain decomposition algorithms discussed above. We are not aware of any domain decomposition algorithm proposed and analyzed for simultaneously a) degen- erate parabolic problems; b) nonlinear and discontinuous transmission conditions; c) Robin and Ventcell transmission conditions; d) global-in-time formulation. Some of these ingredients have appeared previously independently as in [12, 13, 15, 32, 33, 35] and the references therein. The use of higher-order (Ventcell) transmission operators allows physically more valuable information to be exchanged between the subdo- mains and hence typically leads to a better convergence behavior, see [12, 29, 33, 35, 36, 37, 41] and the references therein for linear problems, and [16, 32] for problems with nonlinear reaction terms. The Robin case, obtained by setting one of the Ventcell parameters to zero, is analyzed in [3, 4], both for the exis- tence of a weak solution of the subdomain problem with Robin boundary conditions, and to prove that the space-time DD method is well-defined.

The second purpose of this paper is to derive a posteriori error estimates for the finite volume – backward Euler approximation of the proposed space-time DD algorithm. We build here on the (few) previous contributions on unsteady, nonlinear, and degenerate problems in [20, 23, 42]; we will in particular rely on [23, Theorem 5.2], where the degenerate nature of the problem is handled to obtain an energy-type upper bound on the error. Two additional specific difficulties that were to the best of our knowledge not treated previously are that the current problem features nonlinear and discontinuous transmission conditions and that our finite volume discretization leads to an approximate solution that is nonconforming in space.

Finally, the third purpose of this paper is to address the question of when to stop the domain decom- position iterations, as well as the iterations of the linearization solver used for the subdomain problems. In contrast to prevalent practice, where one iterates until some fixed tolerance has been reached, we propose to stop when the DD/linearization error component is up to a user-specified fraction below the total error.

This typically spares numerous iterations. In practice, this means that the a posteriori error estimates we develop have to hold true at each iteration of the OSWR algorithm and the linearization, and distinguish the different error components (space, time, linearization, and domain decomposition). For this part of our work, we take up the path initiated in [9, 11, 26, 38, 54] for general techniques taking into account inexact algebraic solvers, [45] for (multiscale) mortar techniques, [46, 47] for FETI and BDD domain decomposition algorithms combined with conforming finite element discretizations, and most closely [6, 7] for respectively steady and unsteady linear problems with similar space and time discretizations. To achieve our goals here, we use a) equilibrated H (div)-conforming flux reconstructions, piecewise constant in time, that are direct extension of [7, 25, 54] to our model; b) novel subdomain-wise H ¹ -conforming saturation reconstructions, continuous and piecewise affine in time, which are targeted to the present setting in that they satisfy the nonlinear and discontinuous interface conditions.

The outline of the paper is as follows: Section 2 recalls the physical model and defines weak solutions as

well as the relevant functions spaces. Section 3 presents the space-time domain decomposition method with

nonlinear and discontinuous Ventcell transmission conditions. In Section 4, we present the discrete OSWR

algorithm, by combining a finite volume scheme for the discretization in the individual subdomains and the

backward Euler time stepping with the OSWR method. We then construct the needed ingredients for the

a posteriori error estimates: Section 5 defines the postprocessing as well as the H ¹ - and H (div)-conforming reconstructions. Section 6 puts the pieces together by presenting a guaranteed and fully computable error estimate that bounds the error between the unknown exact solution and the approximate solution. In Section 7 we decompose this overall estimator into individual estimators characterizing the space, time, domain decomposition, and linearization error components. This is used in Section 8 to propose stopping criteria for the OSWR algorithm and for the nonlinear iterations. The method is numerically validated on three examples in three space dimensions in Section 9. Finally, Section 10 draws some conclusions.

2 Presentation of the problem

Let Ω be an open bounded domain of R ^d , d = 2 or 3, which is assumed to be polygonal if d = 2 and polyhedral if d = 3. We denote by ∂Ω its boundary (supposed to be Lipschitz-continuous) and by n the unit normal to ∂Ω, outward to Ω. Let a time interval (0, T ) be given with T > 0. We consider a simplified model of a two-phase flow through a heterogeneous porous medium, in which the advection is neglected.

Assuming that there are only two phases occupying the porous medium Ω, say gas and water, and that each phase is composed of a single component, the mathematical form of this problem as it is presented in [17, 24] is as follows: given initial and boundary gas saturations u 0 and g, as well as a source term f , find u : Ω × [0, T ] → [0, 1] such that

∂ t u − ∇· (λ(u, x) ∇ π(u, x)) = f, in Ω × (0, T ), (2.1a)

u( · , 0) = u 0 , in Ω, (2.1b)

u = g, on ∂Ω × (0, T ). (2.1c)

Here u is the gas saturation (and therefore (1 − u) is the water saturation), π(u, x) : [0, 1] × Ω → R is the capillary pressure , and λ(u, x ) : [0, 1] × Ω → R is the global mobility of the gas given by

λ(u) = λ w (1 − u)λ g (u) λ w (1 − u) + λ g (u) ,

where λ w and λ g are the phase mobilities. One can refer to [20, 21, 24] for a derivation of (2.1a)– (2.1c) from the complete two-phase flow model. For simplicity, we consider only Dirichlet boundary conditions on

∂Ω. Other types of boundary conditions could be dealt with the same way as in [17, 24, 54]. The model problem given by (2.1a) is a nonlinear degenerate parabolic problem as the global mobility λ(u) → 0 for u → 0 and 1, and, moreover, π ⁰ (u) → 0 for u → 0 (see [10, 21]).

2.1 Flow between two rock types

In this part, we particularize the model problem (2.1a) to a porous medium with different capillary pressure curves π i in each subdomain, following [24]. We suppose that Ω is composed of two disjoint subdomains Ω i , i = 1, 2, which are both open polygonal subsets of R ^d with Lipschitz-continuous boundary. We denote by Γ the interface between Ω 1 and Ω 2 , i.e., Γ = (∂Ω 1 ∩ ∂Ω 2 ) ^◦ . Let Γ ^D _i = ∂Ω i ∩ ∂Ω. Both data λ and π, which can in general depend on the physical characteristics of the rock, are henceforth supposed to be homogeneous in each subdomain Ω i , i = 1, 2, i.e., λ i ( · ) := λ |Ω _i ( · ) = λ( · , x), ∀ x ∈ Ω i , and similarly for π i . Equations (2.1a) in each subdomain Ω i then read as

∂ t u i − ∇· (λ i (u i ) ∇ π i (u i )) = f i , in Ω i × (0, T ), (2.2a)

u i ( · , 0) = u 0 , in Ω i , (2.2b)

u i = g i , on Γ ^D _i × (0, T ). (2.2c) We use the notation v i = v | ^Ω i for an arbitrary function v.

Before transcribing the transmission conditions on the interface Γ, we make precise the assumptions on the data (further generalizations are possible, bringing more technicalities):

Assumption 2.1 (Data). 1. For i ∈ { 1, 2 } , π i ∈ C ¹ ([0, 1], R) can be extended in a continuous way to a function (still denoted by π i ) such that π i (u) = π i (0) for all u ≤ 0 and π i (u) = π i (1) for all u ≥ 1.

Moreover, π i | ^[0,1] is a strictly increasing function.

2. For i ∈ { 1, 2 } , λ i ∈ C ⁰ ([0, 1], R ⁺ ) is bounded and can be extended in a continuous way to a function (still denoted by λ i ) such that λ i (u) = λ i (0) for all u ≤ 0 and λ i (u) = λ i (1) for all u ≥ 1 . We denote by C λ an upper bound of λ i (u), u ∈ R .

3. The initial condition is such that u 0 ∈ L ^∞ (Ω) with 0 ≤ u 0 ≤ 1 a.e. in Ω.

4. The boundary conditions 0 ≤ g i ≤ 1 are traces of some functions in L ² (0, T ; H ¹ (Ω i )). For simplicity, we suppose them to be at most piecewise second-order polynomials with respect to the boundary faces of the spatial mesh introduced in Section 4.1.1 below, continuous on Γ ^D _i , and constant in time. Moreover, they need to match in the sense that π 1 (g 1 (x)) = π 2 (g 2 (x)) for all x ∈ Γ ∩ Γ ^D ₁ ∩ Γ ^D ₂ .

5. The source term is such that f ∈ L ² (0, T ; L ² (Ω)). For simplicity we further assume that f is piecewise constant in time with respect to the temporal mesh introduced in Section 4.1.2 below.

For simplicity, we suppose in Assumption 2.1(4) that the boundary conditions are piecewise polynomial in space and constant in time, so that there is in particular no additional data oscillation error stemming therefrom.

We give now the transmission conditions needed to connect the subdomain problems (2.2), for i = 1, 2.

We consider two cases. The first case is when

π 1 (0) = π 2 (0) and π 1 (1) = π 2 (1), (2.3)

the same way as in [17]. If the functions π i satisfy the above condition, the capillarity curves are said to be matching and the resulting transmission conditions on the interface are given by

π 1 (u 1 ) = π 2 (u 2 ), on Γ × (0, T ), (2.4a) λ 1 (u 1 ) ∇ π 1 (u 1 ) · n ₁ = − λ 2 (u 2 ) ∇ π 2 (u 2 ) · n ₂ , on Γ × (0, T ). (2.4b) These conditions yield a discontinuous saturation across the interface, i.e., we find that in general u 1 6 = u 2

on Γ.

In the second case, i.e., in the case when

π 1 (0) 6 = π 2 (0) or π 1 (1) 6 = π 2 (1), (2.5) the capillarity pressure curves are said to be non-matching. Consequently, not only the saturation is discontinuous at the medium interface, but also the capillary pressure field. The condition (2.5), studied in [24], has direct consequences on the behavior of the capillary pressures on both sides of the interface Γ. Indeed, suppose that π 1 (0) ≤ π 2 (0) < π 1 (1) ≤ π 2 (1), that u ^∗ 1 is the unique real in [0, 1] satisfying π 1 (u ^∗ 1 ) = π 2 (0), and that u ^∗ 2 is the unique real in [0, 1] satisfying π 2 (u ^∗ 2 ) = π 1 (1). Then, if u 1 ≥ u ^∗ 1 and u 2 ≤ u ^∗ 2 , we can still prescribe the connection of the capillary pressures on the interface Γ π 1 (u 1 ) = π 2 (u 2 ) as in (2.4a). If 0 ≤ u 1 ≤ u ^∗ 1 , the model imposes u 2 = 0, and the gas phase is entrapped in the rock Ω 1 while the water flows across Γ. In the same way, if u ^∗ ₂ ≤ u 2 ≤ 1, the model prescribes u 1 = 1, and the water phase is captured in Ω 2 as a discontinuous phase while the gas flows across Γ (see Fig. 1 left). Following [21, 24], these conditions on the gas-water saturations on the interface Γ are simply given by

π 1 (u 1 ) = π 2 (u 2 ), on Γ × (0, T ), (2.6a) λ 1 (u 1 ) ∇ π 1 (u 1 ) · n ₁ = − λ 2 (u 2 ) ∇ π 2 (u 2 ) · n ₂ , on Γ × (0, T ), (2.6b) where π i , for i = 1, 2, are truncated capillary pressure functions given on [0, 1] respectively by π 1 : u 7→

max(π 1 (u), π 2 (0)) and π 2 : u 7→ min(π 2 (u), π 1 (1)) (see Fig. 1 right). In [24], it has been established that the model problem (2.2) together with the transmission conditions (2.6) has the necessary mathematical properties to explain the phenomena of gas trapping (see also [14, 8, 19, 52]).

2.2 Transformation of the equations and weak formulation

Still following [24], we present here the mathematical quantities and function spaces used to characterize

the weak solution to the multidomain problem (2.2) with the conditions (2.6). That of the problem (2.2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

1 2 3 4 5 6 7

u

₁

∗ π

₁

( u )

π

₁

(0) π

₁

(1)

u

₁

π

₂

(0)

π

₂

( u ) π

₂

(1)

u

₂

u

^∗₂

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 1 2 3 4 5 6 7

π

₁

( u ) π

₁

(1)

π

₂

( u )

π

₂

(0)

u

₁

∗ u

^∗₂

Figure 1: Capillary pressure curves (left) and truncated capillary pressures curves (right)

with the conditions (2.4) can be deduced straightforwardly from this later, see [17]. As Ω i is a homogeneous rock type, so that π i and λ i do not depend on x, one can define the Kirchhoff transform in each subdomain

ϕ i :

[0, 1] −→ R ⁺ s 7−→ R s

0 λ i (a)π ⁰ _i (a)da. (2.7)

The function ϕ i is Lipschitz-continuous and increasing on [0, 1], and we denote by L ϕ,i its Lipschitz constant, i.e., L ϕ,i := max s∈[0,1] λ i (s)π _i ⁰ (s), so that

| ϕ i (a) − ϕ i (b) | ≤ L ϕ,i | a − b | , ∀ (a, b) ∈ [0, 1] ² ,

and we let L ϕ := max (L ϕ,1 , L ϕ,2 ). We extend the function ϕ i from [0, 1] to R so that ϕ i (u) = ϕ i (0) for all u ≤ 0 and ϕ i (u) = ϕ i (1) for all u ≥ 1.

We now introduce the strictly increasing function φ by

φ :







[π 2 (0), π 1 (1)] −→ R ⁺

s 7−→

Z s π 2 (0)

j∈{1,2} min (λ j ◦ π ⁻¹ _j (a)) da,

and let Π i := φ ◦ π i , for i ∈ { 1, 2 } . The functions Π i | ^[0,1] , introduced first in [24, Lemma 1.2] and used in [4, 3, 17, 19], are differentiable and increasing. We let Π i (u) = Π i (0) for all u ≤ 0 and Π i (u) = Π i (1) for all u ≥ 1 and define the function Π by

Π(u, x) = Π i (u), for x ∈ Ω i .

As Π i are more regular than π i , this allows to connect Π 1 and Π 2 instead of π 1 and π 2 , that is, for all (u 1 , u 2 ) ∈ R ² , we have the crucial equivalence property

π 1 (u 1 ) = π 2 (u 2 ) ⇔ Π 1 (u 1 ) = Π 2 (u 2 ).

Finally, under Assumption 2.1, the function Π i ◦ ϕ ⁻¹ _i is Lipschitz-continuous with a Lipschitz constant lower than 1 (see [24, Lemma 1.2]), i.e.

| Π i (a) − Π i (b) | ≤ | ϕ i (a) − ϕ i (b) | , ∀ (a, b) ∈ [0, 1] ² . (2.8) This inequality will be used in Remark 6.3 below.

We now apply the Kirchhoff transformation (2.7) separately in each subdomain, giving an equivalent formulation suitable for mathematical analysis: find u i such that

∂ t u i − ∆ϕ i (u i ) = f i , in Ω i × (0, T ), (2.9a)

u i ( · , 0) = u 0 , in Ω i , (2.9b)

ϕ i (u i ) = ϕ i (g i ), on Γ ^D i × (0, T ), (2.9c)

together with the conditions at the interface relying on the smoother functions Π i

Π 1 (u 1 ) = Π 2 (u 2 ), on Γ × (0, T ), (2.10a)

∇ ϕ 1 (u 1 ) · n 1 = −∇ ϕ 2 (u 2 ) · n 2 , on Γ × (0, T ). (2.10b) In general, the multidomain problem (2.9)–(2.10) does not have any strong solution, but the introduction of the Kirchhoff transforms ϕ i and of the smoother functions Π i will, following [24], allow for a definition of a weak solution. To this aim, we define

X ϕ _i (g _i ) := L ² (0, T ; H _ϕ ¹ _{i (g i )} (Ω i )),

X Π(g,·) := L ² (0, T ; H _Π(g,·) ¹ (Ω)), X := L ² (0, T ; H ₀ ¹ (Ω)),

where H _ϕ ¹ _{i (g i )} (Ω i ) := { v ∈ H ¹ (Ω i ), v = ϕ i (g i ) on Γ ^D _i } and similarly H _Π(g,·) ¹ (Ω) := { v ∈ H ¹ (Ω), v = Π(g, · ) on ∂Ω } .

We equip the space X with the norm

|| ψ || ^X :=

( Z T

0 ||∇ ψ( · , t) || ² L ² (Ω) dt ) ¹ ₂

. The dual space of X is

X ⁰ := L ² (0, T ; H ⁻¹ (Ω)). (2.11)

We will use the notation h· , ·i H ⁻¹ (Ω),H ₀ ¹ (Ω) to denote the duality pairing between H ⁻¹ (Ω) and H ₀ ¹ (Ω). We will also need the space

Z := H ¹ (0, T ; H ⁻¹ (Ω)).

Using the above developments, a weak solution to problem (2.9)–(2.10) is defined as:

Definition 2.2 (Weak solution). We say that a function u is a weak solution to problem (2.9)–(2.10) if it satisfies:

1. u ∈ Z;

2. u( · , 0) = u 0 ;

3. ϕ i (u i ) ∈ X ϕ _i (g _i ) , where u i := u | ^Ω i , i = 1, 2;

4. Π(u, · ) ∈ X Π(g,·) ;

5. For all ψ ∈ X, the following integral equality holds:

Z T 0

n h ∂ t u, ψ i H ⁻¹ (Ω),H ₀ ¹ (Ω) +

2

X

i=1

( ∇ ϕ i (u i ), ∇ ψ) _{Ω i} − (f, ψ) o

dt = 0. (2.12)

In this paper, we assume that a weak solution given by Definition 2.2 exists. One can then easily show that Π has sufficient regularity to impose the condition (2.10a). Indeed, since Π i ◦ ϕ ⁻¹ _i is a Lipschitz- continuous function, the third point of Definition 2.2 ensures that Π i (u i ) belongs to L ² (0, T ; H ¹ (Ω i )), i = 1, 2. Thus, the point 4 of Definition 2.2 implies the continuity condition (2.10a). Finally, when supposing additionally that the weak solution u is sufficiently regular so that ∂ t u ∈ L ² (0, T ; L ² (Ω)), condition (2.10b) is satisfied weakly using point 5 of Definition 2.2.

Remark 2.3 (Existence and uniqueness of a weak solution) . In [24], the existence of a weak solution to

problem (2.9)–(2.10) (with homogeneous Neumann boundary conditions) was proved using integration by

parts for the time term which requires a stronger test function space. Using the imposed conditions, one also

finds u ∈ L ^∞ (0, T ; L ^∞ (Ω)), with 0 ≤ u ≤ 1 a.e. in Ω × (0, T ). In fact, in [24] (see also [17]), the derived

weak solution is given as the limit of a finite volume approximation of the solution refining the space and time

discretization, and requires a stronger test function space for the application of the Kolmogorov compactness

criterion in L ^∞ . For the uniqueness, a first result was obtained in [17] for the case of matching capillary

pressures curves. For the more general case, the uniqueness is demonstrated only for the one-dimensional

case in [18].

3 Space-time domain decomposition with Ventcell transmission conditions

This section proposes our global-in-time domain decomposition method for the degenerate parabolic prob- lem (2.9) with the nonlinear and discontinuous interface conditions (2.10). We in particular design appro- priate Robin and Ventcell transmission conditions.

An equivalent formulation to the model problem (2.9)–(2.10) can be obtained by solving, for i = 1, 2, equations (2.9) together with optimized Ventcell transmission conditions on Γ × (0, T )

∇ ϕ 1 (u 1 ) · n 1 + L ¹ (Π 1 (u 1 )) = −∇ ϕ 2 (u 2 ) · n 2 + L ¹ (Π 2 (u 2 )), (3.1a)

∇ ϕ 2 (u 2 ) · n 2 + L ² (Π 2 (u 2 )) = −∇ ϕ 1 (u 1 ) · n 1 + L ² (Π 1 (u 1 )), (3.1b) where L ⁱ , i = 1, 2, is a Ventcell (or second-order) boundary operator defined by

L ⁱ (v) := α i,j v + γ i,j (∂ t v − ∆ Γ v), j = (3 − i), (3.2) for a sufficiently regular function v defined on Γ × (0, T ), where ∆ Γ represents the Laplace operator on Γ (the Laplace–Beltrami operator), and α i,j and γ i,j are free parameters that can be chosen so as to optimize the convergence factor of the algorithm (see Remark 3.1 below). A sufficient condition for the equivalence of (2.10) and (3.1) to hold is α i,j > 0 and γ i,j ≥ 0, j = 3 − i, i = 1, 2. The Robin transmission conditions are simply obtained by taking in (3.2) the parameters γ i,j = 0. One can easily show that the operators L ⁱ involve not only the continuity of the function Π as for the Robin case, but also the continuity of the time derivative and of the second-order derivatives of Π along the interface. As a result, this formulation can be seen as a coupling problem between a d-dimensional PDE in the rock Ω i and a (d − 1)-dimensional PDE on the interface Γ between the rock types, which greatly enhances the information exchange between the solutions in the subdomains, see [16, 29, 33, 37] for some model cases.

The Ventcell–OSWR algorithm is defined as follows: the solution u in the whole domain Ω given by u i = u | ^Ω i is approximated by a sequence of solutions u ^k _i , k ≥ 1, defined recursively by solving by

∂ t u ^k _i − ∆ϕ i (u ^k _i ) = f i , in Ω i × (0, T ), (3.3a)

u ^k i ( · , 0) = u 0 , in Ω i , (3.3b)

ϕ i (u ^k i ) = ϕ i (g i ), on Γ ^D i × (0, T ), (3.3c)

∇ ϕ i (u ^k i ) · n _i + L ⁱ (Π i (u ^k i )) = Ψ ^k−1 _i , on Γ × (0, T ), (3.3d) with

Ψ ^k−1 _i := −∇ ϕ j (u ^k−1 _j ) · n _j + L ⁱ (Π j (u ^k−1 _j )), j = (3 − i), k ≥ 2, (3.4) and Ψ ⁰ _i is an initial Ventcell guess on Γ × (0, T ).

A proof of existence of a (suitably defined weak) solution of (3.3) can be found in [4] for the case of Robin boundary conditions.

Remark 3.1 (Optimized parameters). “Optimized parameters” refers to a general way of a priori choosing the Robin or Ventcell parameters α i,j and γ i,j in the OSWR methods so as to numerically minimize the convergence factor for a usually simplified (linearized) problem. Here, these parameters are fixed throughout the OSWR iterations by the optimization as in [29, 33, 37] for domain decomposition in space, in [12, 30, 35, 36, 41] for space-time domain decomposition, and in [16] for semilinear problems. These optimized values are represented by the black boxes in Figure 5 (left) and Figure 10, and we observe that they are not far from the optimal numerical values. The a posteriori analysis and the stopping criteria in this paper are not affected by the specific values of these parameters.

Remark 3.2 (Interface operators). The multi-domain problem (2.9) together with the transmission condi-

tions (3.1) can be formulated through the use of interface operators as a problem posed on the space-time

interface, see [4], where Robin-to-Robin conditions are analyzed, and also [35, 36]. This interface problem

can be solved iteratively by using fixed point iterations (which corresponds to the OSWR algorithm above)

or via a Newton–Krylov method.

Remark 3.3 (Domain decomposition algorithm in physical variables) . The algorithm (3.3) – (3.4) is written in the variables ϕ i (u ^k i ) and Π i (u ^k i ) stemming from the rewriting (2.9) – (2.10) and the corresponding weak formulation of Definition 2.2. One could also formulate a Ventcell–OSWR domain decomposition algorithm directly in the physical variables, using the original formulation (2.2) and the corresponding transmission conditions (2.6):

∂ t u ^k i − ∇· λ i (u ^k i ) ∇ π i (u ^k i )

= f i , in Ω i × (0, T ), (3.5a)

u ^k i ( · , 0) = u 0 , in Ω i , (3.5b)

u ^k _i = g i , on Γ ^D _i × (0, T ), (3.5c) λ i (u ^k _i ) ∇ π i (u ^k _i ) · n i + L ⁱ (u ^k _i ) = Ψ ^k−1 _i , on Γ × (0, T ), (3.5d) with

L ⁱ (u ^k _i ) := α i,j π i (u ^k _i ) + γ i,j ∂ t u ^k _i − ∇ ^Γ · λ i (u ^k _i ) ∇ ^Γ π i (u ^k _i )

, j = (3 − i),

Ψ ^k−1 _i := − λ j (u ^k−1 _j ) ∇ π j (u ^k−1 _j ) · n j + L ⁱ (u ^k _i ), j = (3 − i), k ≥ 2, (3.6) where ∇ ^Γ and ∇ ^Γ · are respectively the surface gradient and divergence on Γ and Ψ ⁰ _i is an initial Ventcell guess on Γ × (0, T ).

Though algorithms (3.3)–(3.4) and (3.5)–(3.6) may differ, an a posteriori error analysis of both algo- rithms can be done simultaneously as in [20], see the “mathematical” scheme of Definition 4.1 and the

“engineering” scheme of Definition 4.4 therein. Indeed, [20, Theorem 3.3] applies to both schemes. We avoid this here for the sake of conciseness.

4 Cell-centered finite volume scheme

We present here a finite volume discretization of the OSWR algorithm (3.3)–(3.4); we also develop lineariza- tion of the subdomain problems by the Newton method, used later in the a posteriori analysis.

4.1 Space-time discretization, notations, and function spaces

We introduce here partitions of Ω and Γ, time discretization, and discrete function spaces.

4.1.1 Partitions of Ω and Γ

Let T ^h,i be a partition of the subdomain Ω i into elements K, such that Ω i = ∪ ^K∈T h,i K; here we suppose that the elements are either simplices or rectangular parallelepipeds but general elements can be treated via submeshes, see [22] and the references therein. Moreover, we assume that the partition is conforming in the sense that if K, L ∈ T ^h,i , K 6 = L, then K ∩ L is either an empty set, a common face, edge, or vertex of K and L.

We then set T ^h = ∪ ² i=1 T ^h,i and denote by h the maximal element diameter in T ^h . The meshes are supposed to be matching on the interface Γ. For all K ∈ T ^h , h K denotes the diameter of the mesh element K and | K | its volume. The interior mesh faces in T ^h,i are collected into the set F h,i ^int ; hence F h,i ^int contains neither the subdomain interfaces nor the outer boundary of Ω. The faces of T ^h,i lying on Γ ^D _i are collected in the set F h,i ^D . We denote by F ^h,i all the faces of T ^h,i and we set F ^h = ∪ ² i=1 F ^h,i . The notation F ^K stands for all the faces of an element K ∈ T ^h . Let F h ^Γ be a partition of Γ given by the faces of T ^h on Γ. We use also E h,i ^int to denote the interior edges (if d = 3, or points if d = 2). We denote by E h,i ^D the edges of F h ^Γ on Γ ^D i . The edges of a face σ ∈ F ^h are collected in the set E ^σ . The volume of a face σ is denoted by | σ | and that of an edge e by | e | . Finally, we use the notation x _K to denote the “center” of the cell K ∈ T ^h . If σ = K | L ∈ F ^h separates the cells K and L, d K,L denotes the Euclidean distance between x _K and x _L , and d K,σ for σ ∈ F ^K denotes the distance from x K to σ. Similarly, we let x σ be the “center” of the face σ and x e the “center” of the edge e and denote respectively by d σ,˜ σ and d σ,e the distance between x σ and x ˜ σ for e = σ | ˜ σ ∈ E h,i ^int and the distance from x σ to e for e ∈ E ^σ .

We assume that the composite mesh T ^h satisfies the following orthogonality condition: for an interface

σ = K | L, the line segment x K x L is orthogonal to this interface (see [27]). The same condition should be

satisfied for an edge e = σ | ˜ σ ∈ E h,i ^int due to the discretization of the Ventcell operator (see Fig. 2).

x

L

σ = K|L

|e|

e = σ|˜ σ d

σ,˜σ

x

σ

x

σ˜

d

K,L

x

σ

x

K

Figure 2: Notation for admissible meshes in three space dimensions

4.1.2 Time discretization

For an integer N ≥ 0, let (τ ⁿ ) _0≤n≤N denote a sequence of positive real numbers corresponding to the discrete time steps such that T = P N

n=1 τ ⁿ . Let t ⁰ = 0, and t ⁿ = P n

j=1 τ ^j , 1 ≤ n ≤ N, be the discrete times. Let I ⁿ = (t ⁿ⁻¹ , t ⁿ ], 1 ≤ n ≤ N . For simplicity, we consider only conforming time grids. The analysis remains valid if we use different time steps in the subdomains, as was done in [7] for a linear diffusion problem.

4.1.3 Notation and discrete function spaces

We denote by P l (S) the space of polynomials on a subdomain S ⊂ Ω of total degree less than or equal to l and by

P l ( T ^h ) := { p h ∈ L ² (Ω); p h | ^K ∈ P l (K), ∀ K ∈ T ^h } ,

the space of piecewise l-degree polynomials on T ^h . We use the notation k·k ^K for the norm in L ² (K), K ∈ T ^h . The corresponding inner product is ( · , · ) K . We also denote by ( · , · ) σ the inner product in L ² (σ) for σ ∈ F ^h . Let | S | be the Lebesgue measure of S, and | σ | the (d − 1)-dimensional Lebesgue measure of σ ∈ R ^d−1 . We define the broken Sobolev space H ¹ ( T ^h ) as the space of all functions v ∈ L ² (Ω) such that v | ^K ∈ H ¹ (K), for all K ∈ T ^h . We will use the sign ∇ to denote the elementwise gradient, i.e., the gradient of a function restricted to a mesh element K ∈ T ^h . We define P _τ ¹ (H ¹ ( T ^h )) and P _τ ¹ (H ₀ ¹ (Ω)) as the space of continuous and piecewise affine in-time functions v hτ with values in H ¹ ( T ^h ) and in H 0 ¹ (Ω), respectively. In both these situations, for every time step 1 ≤ n ≤ N, we will use the abridged notation v _h ⁿ := v hτ ( · , t ⁿ ). By Assumption 2.1, the source function f is piecewise constant in time, i.e., constant on each time interval I ⁿ , 1 ≤ n ≤ N . Its value on I ⁿ is denoted by f ⁿ := f | ^{I n} .

The domain decomposition method we consider is global in time. Thus, at the space-time interface Γ × (0, T ), data should be transferred from one space-time subdomain to the neighboring subdomain.

Then, we denote by P _τ ⁰ (L ² (Γ)) the space of functions piecewise constant in time and with values in L ² (Γ).

Let H(div, Ω) be the space of vector-valued functions from

L ² (Ω) d

that admit a weak divergence in L ² (Ω). Recall that all functions in H(div, Ω) have a continuous normal trace across the interface Γ. As above, P τ ⁰ (H(div, Ω)) is the space of functions piecewise constant in time with values in H(div, Ω). We use RTN 0 (Ω) to denote the lowest-order Raviart–Thomas–N´ed´elec finite-dimensional subspace of H(div, Ω);

any v _h ∈ RTN ₀ (Ω) takes on each element K ∈ T ^h the form [P 0 (K)] ^d +P 0 (K) x for the example of simplices.

Finally, given a function v that is double-valued on the interface Γ, the jump and the average of v across a face σ ∈ F h ^Γ , σ = K | L, K ∈ T ^h,1 , L ∈ T ^h,2 , are defined as

[[v]] = (v K ) | ^σ − (v L ) | ^σ , {{ v }} = (v K ) | ^σ + (v L ) | ^σ

2 .

Here v K = v | ^K is the restriction of v to the element K ∈ T ^h .

4.2 A space-time fully discrete scheme based on finite volumes in space and the backward Euler scheme in time

For the iteration k ≥ 1 of the OSWR algorithm (3.3)–(3.4), subdomain Ω i , i = 1, 2, and the time steps 0 ≤ n ≤ N, let the discrete saturation be identified with the vector of unknowns

(u ^k,n _K ) K∈T h,i , (u ^k,n _K,σ ) σ∈F _K ∩F _h ^Γ ,K∈T _h,i

.

Note that there are in total two interface unknowns u ^k,n _K,σ for each face σ of F h ^Γ , in addition to the usual one unknown u ^k,n _K per mesh element.

We then define the discrete flux F _K,σ ^k,n over a face σ ∈ F ^h ∩ F ^K , K ∈ T ^h,i , by

F _K,σ ^k,n :=



 

 

 

 



 

 

 

 



ϕ i (u ^k,n _K ) − ϕ i (u ^k,n _L ) d K,L

, if σ = K | L ∈ F h,i ^int , ϕ i (u ^k,n _K ) − ϕ i (g i (x σ ))

d K,σ

, if σ ∈ F ^K ∩ F h,i ^D , ϕ i (u ^k,n _K ) − ϕ i (u ^k,n _K,σ )

d K,σ

, if σ ∈ F ^K ∩ F h ^Γ ,

(4.1a)

and the discrete (d − 1)-dimensional flux ¯ F _σ,e ^k,n over an edge e ∈ E ^σ , σ ∈ F ^K ∩ F h ^Γ , K ∈ T ^h,i , by

F ¯ _σ,e ^k,n :=



 

 



 

 



Π i (u ^k,n _K,σ ) − Π i (u ^k,n _{K,˜ σ} ) d σ,˜ σ

, if e = σ | ˜ σ ∈ E ^σ ∩ E h,i ^int , Π i (u ^k,n _K,σ ) − Π i (g i (x e ))

d σ,e

, if e ∈ E ^σ ∩ E h,i ^D .

(4.1b)

We also define the Ventcell term, for K ∈ T ^h,i , Λ ^k,n _K,Γ

σ := α i,j | σ | Π i (u ^k,n _K,σ ) + γ i,j

X

e∈F σ

| e | F ¯ _σ,e ^k,n , σ ∈ F ^K ∩ F h ^Γ , (4.2)

and the information from Ω j , for L ∈ T ^h,j , j = 3 − i, and σ = K | L ∈ F h ^Γ , Ψ ^k−1,n _L,σ := F _L,σ ^k−1,n | σ | + γ i,j

Π j (u ^k−1,n _L,σ ) − Π j (u ^k−1,n−1 _L,σ )

τ ⁿ | σ | +

Λ ^k−1,n _L,Γ

σ . (4.3)

Using (4.1), the approximation of the solution u ^k in the OSWR algorithm (3.3)–(3.4) is: at each iteration k ≥ 1, the initial condition is given by

u ^k,0 _K = 1

| K | (u 0 , 1) K , ∀ K ∈ T ^h , (4.4a)

and for n = 1, . . . , N , the discrete saturations are given by the following finite volume scheme : u ^k,n _K − u ^k,n−1 _K

τ ⁿ | K | + X

σ∈F K

| σ | F _K,σ ^k,n = (f i ⁿ , 1) K ∀ K ∈ T ^h,i , (4.4b)

− F _K,σ ^k,n | σ | + γ i,j

Π i (u ^k,n _K,σ ) − Π i (u ^k,n−1 _K,σ ) τ ⁿ | σ | +

Λ ^k,n _K,Γ

σ = Ψ ^k−1,n _L,σ , ∀ σ = K | L ∈ F h ^Γ . (4.4c)

4.3 Newton linearization

At Newton linearization step m ≥ 1 the following problem is to be solved:

For i = 1, 2, find

(u ^k,n,m _K ) K∈T _h,i , (u ^k,n,m _K,σ ) _{σ∈F K ∩F} ^Γ

h ,K∈T h,i

∈ P 0 ( T ^h,i ) × P 0 ( F h ^Γ ) such that u ^k,n,m _K − u ^k,n−1 _K

τ ⁿ | K | + X

σ∈F K

| σ | F _K,σ ^k,n,m−1 = (f i ⁿ , 1) K , ∀ K ∈ T ^h,i , (4.5a)

−| σ | F _K,σ ^k,n,m−1 + γ i,j

Π ^m−1 _i (u ^k,n,m _K,σ ) − Π i (u ^k,n−1 _K,σ )

τ ⁿ | σ | +

Λ ^k,n,m−1 _K,Γ

σ = Ψ ^k−1,n _L,σ , ∀ σ = K | L ∈ F h ^Γ , (4.5b) where F _K,σ ^k,n,m−1 are the linearized face fluxes and Λ ^k,n,m−1 _K,Γ are the linearized Ventcell terms (affine functions of u ^k,n,m _K and u ^k,n,m _K,σ obtained by linearization around the available u ^k,n,m−1 _K and u ^k,n,m−1 _K,σ , see [1] for details), and the function Π i is linearized by

Π ^m−1 _i (u ^k,n,m _K,σ ) := Π i (u ^k,n,m−1 _K,σ ) + ∂Π i

∂u K,σ

(u ^k,n,m−1 _K,σ ) · (u ^k,n,m _K,σ − u ^k,n,m−1 _K,σ ).

Equations (4.5a)–(4.5b) form a system of linear algebraic equations that is solved at each domain de- composition iteration k, time step n, and Newton iteration m, independently in each of the subdomains Ω i .

5 Postprocessing and H ¹ - and H(div)-conforming reconstructions

So far, we have described the OSWR domain decomposition algorithm and its finite volume discretization.

In the next four sections, we address the important question of assessing the accuracy of the approximate solution, both as a function of the space and time discretization parameters, and as a function of the OSWR and Newton iterations. This is achieved by deriving detailed a posteriori error estimators for each of the possible error components.

At each OSWR iteration k ≥ 1, time step 0 ≤ n ≤ N , and linearization step m ≥ 1, we dispose of the piecewise constant in space and in time approximation of the saturation given by the values u ^k,n,m _K , K ∈ T ^h , defined in (4.5). In this section, we develop basic tools that will allow us to build the estimators.

We extend for this purpose to the present setting the approach of [25, 45, 54, 23] and the references therein and proceed in three steps:

(i) Define a locally postprocessed saturation u ^k,m _hτ that is elementwise more regular than the piecewise constants u ^k,n,m _K , K ∈ T ^h . This is customary in finite volume methods, as an energy-norm-type a posteriori error estimate has no meaning for the piecewise constants. There, however, holds (recall that we use the notation u ^k,m _hτ,i = u ^k,m _hτ | ^Ω i )

ϕ i (u ^k,m _hτ,i ) 6∈ X ϕ _i (g _i ) , i = 1, 2, and Π 1 (u ^k,m _hτ,1 ) 6 = Π 2 (u ^k,m _hτ,2 ) on Γ, (5.1) since u ^k,m _hτ is discontinuous between mesh elements. Thus, u ^k,m _hτ is a nonconforming approximation of the weak solution of Definition 2.2 that has to in particular satisfy properties 3 and 4.

(ii) Define a saturation reconstruction s ^k,m _hτ that is conforming in the sense of Definition 2.2, satisfying all s ^k,m _hτ ∈ Z, ϕ i (s ^k,m _hτ,i ) ∈ X ϕ i (g i ) , i = 1, 2, Π(s ^k,m _hτ , · ) ∈ X Π(g,·) . (5.2) This is a first crucial ingredient of our a posteriori error estimate and plays the role of the potential reconstruction in terms of [25]. It will be constructed cheaply, in an explicit way by local averaging.

(iii) Define an equilibrated flux reconstruction σ ^k,m _hτ (in terms of [25]), piecewise constant in-time and H(div, Ω)-conforming, locally conservative on the mesh T ^h :

σ ^k,n,m _h := σ ^k,m _hτ | ^{I n} ∈ RTN ₀ (Ω), (5.3a)

f ⁿ − u ^k,n,m _K − u ^k,n−1 _K

τ ⁿ − ∇· σ ^k,n,m _h , 1

!

K

= 0, ∀ K ∈ T ^h . (5.3b)

This is again constructed cheaply, by solving a coarse problem at each time step followed by some

local Neumann problems in bands close to the interface.

5.1 Postprocessed saturation u ^k,m _hτ

Consider an OSWR iteration k ≥ 1, time step 1 ≤ n ≤ N , and a linearization step m ≥ 1.

Extending [28, 25] to the present nonlinear context, we define the postprocessed approximation ϕ ^k,n,m _h,i ∈ P 2 ( T ^h,i ) such that ∇ ϕ ^k,n,m _h,i ∈ RTN ₀ (K) in each element K ∈ T ^h,i , as the solution of

( −∇ ϕ ^k,n,m _h,i | ^K · n i ) | ^σ = F _K,σ ^k,n,m−1 ∀ σ ∈ F ^K , (5.4a) (ϕ ⁻¹ _i (ϕ ^k,n,m _h,i ), 1) K

| K | = u ^k,n,m _K . (5.4b)

Procedure (5.4) is local in each mesh element and its cost is negligible. From ϕ ^k,n,m _h,i , we construct the corresponding continuous, piecewise affine in-time postprocessed saturation by

u ^k,m _hτ,i ( · , t n ) = ϕ ⁻¹ _i (ϕ ^k,n,m _h,i ), 1 ≤ n ≤ N. (5.5) Note that u ^k,m _hτ is nonconforming in the sense of (5.1).

Remark 5.1 (Practice). In practice, condition (5.4b) is typically approximated by the quadrature rule ϕ ^k,n,m _h,i ( x _K ) = ϕ i (u ^k,n,m _K ) ∀ K ∈ T ^h,i , so that ϕ ^k,n,m _h,i becomes readily available. Moreover, the significant estimators in Theorem 6.1 below only involve ϕ ^k,n,m _h,i and not u ^k,m _hτ,i , which then needs not be constructed in practice.

5.2 Subdomain H ¹ -conforming reconstruction s ^k,m _hτ

We define in this section the saturation reconstruction s ^k,m _hτ . It will belong to H ¹ (Ω i ), i = 1, 2, and satisfy (5.2) (up to numerical quadrature). In contrast to the usual scenario, see [25, 45, 54], we cannot immediately apply the operator I ^av that prescribes averages at the Lagrange degrees of freedom to the postprocessed saturation u ^k,m _hτ constructed in Section 5.1, since u ^k,m _hτ is generally not a piecewise polynomial and, moreover, it may not be constructed in practice, see Remark 5.1. Instead, we proceed in two steps.

First, from the available ϕ ^k,n,m _h,i ∈ P 2 ( T ^h,i ) defined by (5.4), k ≥ 1, 1 ≤ n ≤ N , m ≥ 1, we define a piecewise polynomial that is continuous in each subdomain Ω i by

I ^av (ϕ ^k,n,m _h,i ).

Here I ^av : P 2 ( T ^h,i ) → P 2 ( T ^h,i ) ∩ H ¹ (Ω i ) is the averaging operator given by I ^av (φ h )(x) = 1

|T ^x | X

K∈T _x

φ h | ^K (x),

with T ^x the set of all the elements of T ^h,i sharing the Lagrange degree of freedom x, located in the interior of Ω i or at the interface Γ. Second, we consider ϕ ⁻¹ _i ( I ^av (ϕ ^k,n,m _h,i )(x)) at the Lagrange degrees of freedom x to define s ^k,n,m _h ; we modify it at the interface Γ, on the boundaries Γ ^D i , and in the element interiors to satisfy

Π 1 (s ^k,n,m _h | ^Ω 1 ) = Π 2 (s ^k,n,m _h | ^Ω 2 ) on Γ, (5.6a) s ^k,n,m _h | Γ ^D _i = g i on Γ ^D _i , i = 1, 2, (5.6b) 1

| K | (s ^k,n,m _h , 1) K = u ^k,n,m _K , ∀ K ∈ T ^h . (5.6c) Finally, the continuous piecewise affine in-time function s ^k,m _hτ is prescribed by

s ^k,m _hτ ( · , t n ) = s ^k,n,m _h , 1 ≤ n ≤ N. (5.7) Remark 5.2 (Practice). In practice, condition (5.6a) will again only be satisfied in the sense of quadrature.

The last condition (5.6c) is easy to satisfy using the bubble functions following [25, Section 3.2.2], again up

to a quadrature error.

5.3 Equilibrated flux reconstruction σ ^k,m _hτ

Because of the domain decomposition formulation with Robin or Ventcell transmission conditions, the finite volume fluxes F _K,σ ^k,n of (4.1a) (or their linearizations F _K,σ ^k,n,m−1 ) do not match across the two sides of the interface Γ. Consequently, −∇ ϕ ^k,n,m _h is not H(div, Ω)-conforming (it does not lie in the space RTN ₀ (Ω)), and cannot be used as a flux reconstruction in the sense of (5.3). To obtain σ ^k,m _hτ , the procedure used in the linear case in [6, 7] can be employed here. The details are given in Appendix A.

6 A posteriori error estimate

Relying on the developments of the previous sections, we derive here an a posteriori error estimate that bounds an energy error between the postprocessed saturation u ^k,m _hτ defined in Section 5.1 and the weak solution u of Definition 2.2 by a guaranteed and fully computable upper bound, and this at each OSWR iteration k ≥ 1 and at each linearization step m ≥ 1.

For all times t ∈ (0, T ), let

Q t,i := L ² (0, t; L ² (Ω i )), X t := L ² (0, t; H ₀ ¹ (Ω)), X _t ⁰ := L ² (0, t; H ⁻¹ (Ω)).

Because of the assumptions on the weak solution u in Definition 2.2 and since, by (5.5), ϕ i (u ^k,m _hτ,i ) ∈ L ² (0, T ; L ² (Ω i )) for the postprocessed saturation u ^k,m _hτ defined in Section 5.1, u ^k,m _hτ ∈ X ⁰ , and u ^k,m _hτ ( · , T ) ∈ H ⁻¹ (Ω), we can define an energy-type error following [23] as

k u − u ^k,m _hτ k ² ]

:=

2

X

i=1

k ϕ i (u i ) − ϕ i (u ^k,m _hτ,i ) k ² Q T ,i + L ϕ

2 k u − u ^k,m _hτ k ² X ⁰ + L ϕ

2 k (u − u ^k,m _hτ )( · , T ) k ² H ⁻¹ (Ω)

+ 2

2

X

i=1

Z T

0 k ϕ i (u i ) − ϕ i (u ^k,m _hτ,i ) k ² Q t,i + Z t

0 k ϕ i (u i ) − ϕ i (u ^k,m _hτ,i ) k ² Q s,i e ^t−s ds

! dt,

(6.1)

where we recall that L ϕ is the maximal Lipschitz constant of the functions ϕ i defined by (2.7).

The saturation reconstruction s ^k,m _hτ defined in Section 5.2 is conforming in the sense of Definition 2.2 in that it satisfies (5.2). Consequently, it can be plugged into the weak form (2.12) to yield the residual

hR (s ^k,m _hτ ), ψ i ^{X 0 ,X} :=

Z T 0

n (f, ψ) − h ∂ t s ^k,m _hτ , ψ i ^{H −1 (Ω),H} 0 ¹ (Ω) −

2

X

i=1

( ∇ ϕ i (s ^k,m _hτ,i ), ∇ ψ) Ω _i

o (θ)dθ. (6.2)

The dual norm of the residual R (s ^k,m _hτ ) on X ⁰ is then given by kR (s ^k,m _hτ ) k ^{X 0} := sup

ψ∈X, kψk X =1 hR (s ^k,m _hτ ), ψ i ^{X 0 ,X} (6.3) and expresses by how much s ^k,m _hτ fails to satisfy (2.12); we will also use similar notation for X _t ⁰ , where T is replaced by t. We finally define the weaker error

k u − s ^k,m _hτ k ^[ :=

r L ϕ

2 (

(2e ^T − 1) k u 0 − s ^k,m _hτ ( · , 0) k ² H ⁻¹ (Ω) + kR (s ^k,m _hτ ) k ² X ⁰

+ 2 Z T

0

kR (s ^k,m _hτ ) k ² X _t ⁰ + Z t

0 kR (s ^k,m _hτ ) k ² X _s ⁰ e ^t−s ds dt

) ¹ ₂ .

(6.4)

For the iteration k ≥ 1 of the OSWR algorithm, for a linearization step m ≥ 1, and for a time step 1 ≤ n ≤ N, we define the initial condition, residual, and discretization estimators respectively by:

η _IC ^k,m := k u 0 − s ^k,m _hτ ( · , 0) k ^{H −1 (Ω)} , (6.5a)

η _R,K ^k,n,m := C P,K h K k f ⁿ − ∂ t s ^k,m _hτ − ∇· σ ^k,n,m _h k ^K , K ∈ T ^h , (6.5b)

η _disc,K,i ^k,n,m (t) := k σ ^k,n,m _h + ∇ ϕ i (s ^k,m _hτ ( · , t)) k ^K , K ∈ T ^h,i , t ∈ I ⁿ , i = 1, 2, (6.5c)

where, in (6.5b), C P,K is the constant in the Poincar´e inequality:

k q − q K k ≤ C P,K h K k∇ q k ^K ∀ q ∈ H ¹ (K),

with q K the mean value of the function q on the element K and C P,K = 1/π whenever K is convex. Then the estimators η ^k,n,m and η ^k,m are respectively defined by

η ^k,n,m :=





 Z

I ⁿ 2

X

i=1

X

K∈T _h,i

η _R,K ^k,n,m + η ^k,n,m _disc,K,i (t) 2

dt







1 2

, 1 ≤ n ≤ N, (6.5d)

η ^k,m :=

r L ϕ

2 (

(2e ^T − 1) η _IC ^k,m 2

+

N

X

n=1

η ^{k,n,m 2}

+ 2

N

X

n=1

τ ⁿ

n

X

l=1

η ^k,l,m 2

+ 2

N

X

n=1 n

X

l=1

Z

I ⁿ

Z

I ^l

e ^t−s dsdt X ^l

q=1

η ^{k,q,m 2} ) ¹ ₂

.

(6.5e)

Our main result is stated in the following theorem:

Theorem 6.1 (A posteriori error estimate). Let u be the weak solution of the multidomain problem (2.9)–

(2.10) in the sense of Definition 2.2. Consider the Ventcell–OSWR algorithm (3.3)–(3.4) at iteration k ≥ 1, the finite volume discretization of Section 4.2 on space and time meshes indexed by h and τ, and the Newton linearization of Section 4.3 on step m ≥ 1. Let u ^k,m _hτ be the postprocessed saturation of Section 5.1, s ^k,m _hτ the saturation reconstruction of Section 5.2, and σ ^k,m _hτ the equilibrated flux reconstruction of Section 5.3. Then

k u − s ^k,m _hτ k ^[ ≤ η ^k,m . (6.6)

Moreover, if

¯

ϕ ∈ X, where ϕ ¯ | ^Ω i := ϕ i (u i ) − ϕ i (s ^k,m _hτ,i ), i = 1, 2, (6.7) then

k u − u ^k,m _hτ k ^] ≤ η ^k,m + k u ^k,m _hτ − s ^k,m _hτ k ^] . (6.8) Proof. We have defined s ^k,m _hτ so that ϕ i (s ^k,m _hτ,i ) ∈ X ϕ i (g i ) , Π(s ^k,m _hτ , · ) ∈ X Π(g,·) , and

(f ⁿ − ∂ t s ^k,m _hτ | ^{I n} − ∇· σ ^k,n,m _h , 1) K = 0 ∀ K ∈ T ^h , 1 ≤ n ≤ N. (6.9) Indeed, (6.9) follows from the requirement (5.6c) which implies

(∂ t s ^k,m _hτ | ^{I n} , 1) K = s ^k,n,m _h − s ^k,n−1 _h τ ⁿ , 1

!

K

= u ^k,n,m _K − u ^k,n−1 _K τ ⁿ , 1

!

K

together with (5.3b). Thus, adding and subtracting ( σ ^k,m _hτ , ∇ ψ) in (6.2) and using Green’s theorem and the fact that ∂ t s ^k,m _hτ ∈ L ² (0, T ; L ² (Ω)), we infer

hR (s ^k,m _hτ ), ψ i ^{X 0 ,X}

=

N

X

n=1

Z

I ⁿ

n (f − ∂ t s ^k,m _hτ − ∇· σ ^k,n,m _h , ψ) −

2

X

i=1

( ∇ ϕ i (s ^k,m _hτ,i ) + σ ^k,n,m _h , ∇ ψ) Ω _i

o (θ)dθ.

From (6.3) and using (6.9), it then immediately follows that kR (s ^k,m _hτ ) k ² X ⁰ ≤

N

X

n=1

η ^{k,n,m 2} .

Proceeding as in [23, Theorem 5.3], we then promptly arrive at (6.6). Next, it follows by inspection of the proof of [23, Theorem 5.2] that, under assumption (6.7), we have

k u − s ^k,m _hτ k ^] ≤ k u − s ^k,m _hτ k ^[ . (6.10)

Thus (6.8) follows by the triangle inequality

k u − u ^k,m _hτ k ^] ≤ k u − s ^k,m _hτ k ^] + k u ^k,m _hτ − s ^k,m _hτ k ^] .

Remark 6.2 (Condition (6.7)). Condition (6.7) seems unfortunately necessary to apply [23, Theorem 5.2]

so as to obtain (6.10) and consequently the a posteriori estimate (6.8). It is indeed rather restrictive for the complete problem (2.9) – (2.10) . It is, though, in particular satisfied when the global mobilities of the gas λ i

are constants in the respective subdomains Ω i and π 1 (0) = π 2 (0) . In this case, we get an estimate in the norm k·k ^] defined by (6.1) for the present nonlinear degenerate parabolic problem, where the transmission conditions are nonlinear and indeed discontinuous, since π 1 (u 1 ) = π 2 (u 2 ) but u 1 6 = u 2 in general at the interface Γ. In general, when condition (6.7) is not satisfied, we only control the dual residual norm k·k ^[ defined by (6.4).

Remark 6.3 (A posteriori estimate of the saturation and capillary pressure errors). Applying (2.8), the above estimators bound also the saturation and the capillary pressure errors. More precisely, the estimates still hold if we replace the functions ϕ i by the functions Π i in (6.1).

7 Distinguishing the space, time, linearization, and the DD errors

In this section, we distinguish the different error components, proceeding as in [7, 20, 23, 26] and the references therein. The aim is in particular to separate the domain decomposition error from the estimated space, time, and linearization errors.

For the iteration k ≥ 1 of the OSWR algorithm, for all time steps 1 ≤ n ≤ N , a linearization step m ≥ 1, and both subdomains Ω i , i = 1, 2, we define a vector-valued function ` ^k,n,m _h,i ∈ RTN(Ω i ) that approximates the available flux used in the Newton iterations in Section 4.3, i.e.,

(` ^k,n,m _h,i · n i ) | ^σ = F _K,σ ^k,n,m ∀ σ ∈ F ^K , ∀ K ∈ T ^h,i , (7.1) where the fluxes F _K,σ ^k,n,m are obtained by plugging the available u ^k,n,m _K to the nonlinear flux expression (4.1a).

The vector function ` ^k,n,m _h,i is called the linearized flux. It tends to −∇ ϕ ^k,n,m _h,i defined in (5.4a) at convergence of the Newton algorithm. For all K ∈ T ^h,i , we then define the local spatial, temporal, domain decomposition, and linearization estimators by:

η ^k,n,m _sp,K,i := η ^k,n,m _R,K + k∇ ϕ i (s ^k,m _hτ ( · , t n )) + ` ^k,n,m _h,i k ^K , (7.2a) η ^k,n,m _tm,K,i (t) := k∇ ϕ i (s ^k,m _hτ ( · , t)) − ϕ i (s ^k,m _hτ ( · , t n ))

k ^K , (7.2b)

η _dd,K,i ^k,n,m := k∇ ϕ ^k,n,m _h,i + σ ^k,n,m _h k ^K , (7.2c) η ^k,n,m _lin,K,i := k∇ ϕ ^k,n,m _h,i + ` ^k,n,m _h,i k ^K . (7.2d) Note that (7.2c) and (7.2d) only work with lowest-order Raviart–Thomas–N´ed´elec polynomials, so that they can be evaluated without numerical quadrature error and fast. Set, like in (6.5d), for a = sp, tm, dd, lin,

η _a,i ^{k,n,m 2} :=

Z

I ⁿ

X

K∈T _h,i

η _a,K,i ^{k,n,m 2}

dt and η ^k,n,m _a ² :=

2

X

i=1

η ^k,n,m _a,i ²

, (7.3)

and note that except for a = tm, η ^k,n,m _a,i 2

= τ ⁿ P

K∈T h,i η _a,K,i ^k,n,m 2

; for a = tm, the dependence of the estimators on time is left implicit. The global versions, like in (6.5e) but without the initial condition, are

η a ^k,m :=

r L ϕ

2 ( _N

X

n=1

η a ^k,n,m

² ) ¹ ₂

+ √ 2

( _N X

n=1

τ ⁿ

n

X

l=1

η a ^k,l,m

² ) ¹ ₂

+ √ 2

( _N X

n=1 n

X

l=1

Z

I ⁿ

Z

I ^l

e ^t−s dsdt ^l

X

q=1

η _a ^k,q,m 2 ) ¹ ₂ !

+ δ a k u ^k,m _hτ − s ^k,m _hτ k ^] ,

(7.4)

where δ a = 0 for a = tm, dd, lin and δ sp will take the values 0 or 1, depending on the norm in which the error is measured. Then using estimates (6.8) and (6.6) together with the triangle inequality gives:

Corollary 7.1 (A posteriori error estimate distinguishing error components). Let the assumptions of The- orem 6.1 be satisfied. Let the linearized flux ` ^k,n,m _h be given by (7.1) and the estimators by (6.5a) and (7.2)–

(7.4). Then, with δ sp = 0,

k u − s ^k,m _hτ k ^[ ≤ r L ϕ

2

p 2e ^T − 1η _IC ^k,m + η ^k,m sp + η ^k,m tm + η _dd ^k,m + η _lin ^k,m , and, under condition (6.7) and with δ sp = 1,

k u − u ^k,m _hτ k ^] ≤ r L ϕ

2

p 2e ^T − 1η ^k,m _IC + η ^k,m sp + η tm ^k,m + η _dd ^k,m + η ^k,m _lin .

Remark 7.2 (Error components distinction). The distinction in Corollary 7.1 is merely heuristic, as we do not currently have a proof that our estimates on the individual error components are reliable and efficient bounds on the corresponding error components, i.e., that they represent, up to a constant, their upper and lower bounds. This seems currently out of reach for the present highly nonlinear unsteady problem. The numerical experiments in Section 9 below, though, suggest that η _dd ^k,m is indeed an upper bound on the domain discretization error (cf. Figure 5, right), and each component estimate is essentially only sensitive to the corresponding error component (for example, in Figure 5, right, essentially only η ^k,m _dd evolves during the DD iterations, whereas in Figure 8, essentially only η _lin ^k,m evolves during the linearization iterations). Finer estimates with reliable and efficient separate bounds for the space discretization and linear algebra error components were recently obtained in a simpler setting of a model steady linear problem in [43, 44].

8 Stopping criteria and optimal balancing of the different error components

We provide here stopping criteria for the OSWR algorithm and the nonlinear solver for the subdomain problems as in [7, 20, 23, 26] and the references therein. The goal is to efficiently use the computer resources while saving many unnecessary iterations and to equilibrate all error components.

Let two real parameters δ lin and δ dd be given in (0, 1). The stopping criteria for the linearization step (inner loop in m) in each subdomain i, at each time step n, and each OSWR iteration k is chosen as, employing the estimates (7.3),

η _lin,i ^k,n,m ≤ δ lin max n

η _sp,i ^k,n,m , η _tm,i ^k,n,m , η _dd,i ^k,n,m o

, i = 1, 2. (8.1)

Similarly, the stopping criteria for the OSWR algorithm (outer loop in k) is set as, employing the esti- mates (7.4),

η _dd ^k,m ≤ δ dd max n

η _sp ^k,m , η _tm ^k,m o

. (8.2)

The first criterion (8.1) stipulates that there is no need to continue with the linearization iterations if

the overall error is dominated by one of the other components. That of the second criterion (8.2) decides

that we stop the OSWR algorithm if the domain decomposition error is dominated by one of the other

components. The entire procedure of the approach is then described by the following adaptive algorithm: